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The Creation-Discovery-View : Towards a Possible Explanation of Quantum Reality

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The creation-discovery-view : towards a possible explanation of quantum reality¤ Diederik Aerts and Bob Coecke CLEA, Free University of Brussels, Pleinlaan 2, 1050 Brussels, Belgium. e-mails: [email protected], [email protected] Abstract We present a realistic interpretation for quantum mechanics that we have called the 'creation discovery view' and that is being developed in our group in Brussels. In this view the change of state of a quantum entity during an experiment is taken to be a 'real change' under inuence of the experiment, and the quantum probability that corresponds to the experi- ment is explained as due to a lack of knowledge of a deeper deterministic reality of the measurement process. The technical mathematical theory underlying the creation discovery view that we are elaborating we have called the 'hidden measurement formalism'. We present a simple physical example: the 'quantum machine', where we can illustrate easily how the quantum structure arises as a consequence of the two mentioned e®ects, a real change of the state, and a lack of knowledge about a deeper reality of the measurement process. We analyze non-locality in the light of the creation discovery view, and show that we can understand it if we accept that also the basic concept of 'space' is partly due to a creation: when a detection of a quantum entity in a non-local state occurs, the physical act of detection itself 'creates' partly the 'place' of the quantum entity. In this way the creation discovery view introduces a new ontology for space: space is not the all embracing theater, where all 'real' objects have their place, but it is the structure that governs a special type of relations (the space-like relations) between macroscopic physical entities. We bring for- ward a number of elements that show the plausibility of the approach and also analyze the way in which the presence of Bell-type correlated events can be incorporated. ¤Published as: Aerts, D. and Coecke, B., 1999, \The creation-discovery-view : towards a possible explanation of quantum reality", in Language, Quantum, Music, eds. Dalla Chiara, M.L., Kluwer Academic, Dordrecht. 1 1 Introduction The creation discovery view and together with it its technically underlying hid- den measurement formalism has been elaborated from the early eighties on, and many aspects of it have been exposed in di®erent places [6, 7, 12, 13, 15, 16, 19, 20, 22, 23, 30, 31, 32, 33, 34, 35, 36]. In this paper we give an overview of the most important of these aspects. Quantum mechanics was originally introduced as a non-commutative ma- trix calculus of observables by Werner Heisenberg [41] and parallel as a wave mechanics by Erwin SchrÄodinger [43]. These two structurally very di®erent theories could explain fruitfully the early observed quantum phenomena. Al- ready in the same year the two theories were shown to be realizations of the same, more abstract, ket-bra formalism by Dirac [38]. Only some years later, in 1934, John Von Neumann put forward a rigorous mathematical framework for quantum theory in an in¯nite dimensional separable complex Hilbert space [46]. Matrix mechanics and wave mechanics appear as concrete realizations: the ¯rst one if the Hilbert space is l2, the collection of all square summable complex numbers, and the second one if the Hilbert space is L2, the collection of all square integrable complex functions. The formulation of quantum mechanics in the abstract framework of a complex Hilbert space is now usually referred to as the 'standard quantum mechanics'. The basic concepts - the vectors of the Hilbert space representing the states of the system and the self-adjoint operators representing the observables - in this standard quantum mechanics are abstract mathematical concepts de¯ned mathematically in and abstract mathematical space. Several approaches have generalized the standard theory starting from more physically de¯ned basic concepts. John Von Neumann and Garett Birkho® have initiated one of these approaches [29] were they analyze the di®erence between quantum and classical theories by studying the 'experimental propositions'. They could show that for a given physical system classical theories have a Boolean lattice of experimental propositions while for quantum theory the lattice of experimental propositions is not Boolean. Similar fundamental structural di®erences between the two theories have been investigated by concentrating on di®erent basic concepts. The collection of observables of a classical theory was shown to be a commutative algebra while this is not the case for the collection of quantum observables [40, 44]. Luigi Accardi and Itamar Pitowski obtained an analogous result by concentrating on the probability models connected to the two theories: classical theories have a Kolmogorovian probability model while the probability model of a quantum theory is non Kolmogorovian [1, 42]. The fundamental structural di®erences between the two types of theories, quantum and classical, in di®erent categories, was interpreted as indicating also a fundamental di®erence on the level of the nature of the reality that both theories describe: the micro world should be 'very di®erent' from the macro world. This state of a®airs was very convincing also because concrete attempts 2 to understand quantum mechanics in a classical way had failed as well: e.g. the many 'physical' hidden variable theories that had been tried out [45]. The structural di®erence between quantum theories and classical theories (Boolean lattice versus non- Boolean lattice of propositions, commutative algebra ver- sus non commutative algebra of observables and Kolmogorovian versus non Kolmogorovian probability structure) had been investigated mostly mathemat- ically and not much understanding of the physical meaning of the structural di®erences had been gained during all these years. The ¯rst step that led to the creation discovery view and its underlying hidden measurement formalism was a breakthrough in the understanding of the physical origin of these mathematical structural di®erences between quantum and classical theories. Indeed, one of the authors found in the early eighties a way to identify the physical aspects that are at the origin of the structural di®erences [3, 6, 7]. Let us summarize these ¯ndings: it are mainly two aspects that determine the mathematical structural di®erences between classical and quantum theories in the di®erent categories: We have a quantum-like theory describing a system under investigation if the measurements needed to test the properties of the system are such that: (1) The measurements are not just observations but provoke a real change of the state of the system. (2) There exists a lack of knowledge about the reality of what happens during the measurement process. The presence of these two aspects is su±cient to render the description of the system under consideration quantum-like. It is the lack of knowledge (2) that is theoretically structured in a non Kolmogorovian probability model. In a certain sense it is possible to interpret the second aspect, the presence of the lack of knowledge on the reality of the measurement process, as the presence of 'hidden measurements' instead of 'hidden variables'. Indeed, if a measurement is performed with the presence of such a lack of knowledge, then this is actually the classical mixture of a set of classical hidden measurements, were for such a classical hidden measurement there would be no lack of knowledge. In an analogous way as in a hidden variable theory, the quantum state is a classical mixture of classical states. This is the reason why we have called the underlying theory of the creation discovery view the hidden measurement formalism. It is possible to illustrate the creation discovery view and the hidden measurement aspect in a very simple way by using a mechanical model that was introduced in [5, 16, 7] and that we have called the quantum machine. This is the subject of next section. 3 2 The Quantum Machine. Several aspects of the quantum machine have been presented in di®erent occa- sions [6, 7, 8, 9, 10, 15, 19, 20] and we shall therefore introduce here only the basic aspects. The machine that we consider consists of a physical entity S that is a point particle P that can move on the surface of a sphere, denoted surf, with center O and radius 1. The unit-vector v where the particle is located on surf represents the state pv of the particle (see Fig 1,a). For each point u surf, we introduce the following measurement eu. We consider the diamet- rically2 opposite point u, and install a piece of elastic of length 2, such that it is ¯xed with one of its¡end-points in u and the other end-point in u. Once the elastic is installed, the particle P falls from its original place v orthogonally¡ onto the elastic, and sticks on it (Fig 1,b). Then the elastic breaks and the particle P , attached to one of the two pieces of the elastic (Fig 1,c), moves to one of the two end-points u or u (Fig 1,d). Depending on whether the particle P arrives ¡ u u in u (as in Fig 1) or in u, we give the outcome o1 or o2 to eu. We can easily calculate the probabilities¡ corresponding to the two possible outcomes. v v P u u Fig 1 : A representation of the quantum machine. q q In (a) the physical entity P is in state pv in the P point v, and the elastic corresponding to the mea- surement eu is installed between the two diametri- -u -u cally opposed points u and u. In (b) the particle (a) ¡ (b) P falls orthogonally onto the elastic and stick to u u it.
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